Nuclear magnetic resonance measurement techniques in non-uniform fields

ABSTRACT

Methods and pulse sequences for facilitating nuclear magnetic resonance (NMR) measurements in grossly inhomogeneous fields. Methods and pulse sequences according to the invention may be used to accurately measure variables such as transverse relaxation time, longitudinal relaxation time, and diffusion, without the need for data at long recovery time, thereby allowing for faster measurements. In addition, methods and pulse sequences according to embodiment of the invention may allow simultaneous encoding of information in both the amplitude and the shape of echoes, so as to allow a single-shot measurement of multiple variables, e.g., both transverse relaxation time (from the decay of echo amplitudes) and longitudinal relaxation time (from the echo shape). CPMG detection may be used to overcome the often limited signal-to-noise ratio in grossly inhomogeneous fields.

BACKGROUND

1. Field of Invention

The present invention relates to nuclear magnetic resonance techniques for measuring parameters of a sample, particularly in inhomogeneous magnetic fields.

2. Discussion of Related Art

In standard nuclear magnetic resonance (NMR) spectroscopy, the primary information of interest is contained in the spectrum of the signal. This is made possible because magnets are now available with homogeneities typically better than 1 part in 10⁸. However, some applications require large samples which are unable to fit inside standard superconducting magnets, and thus require the use of one-sided magnet systems. As a result, the magnetic field across these samples is necessarily inhomogeneous and the signal-to-noise ratio is intrinsically low. One such application is in the field of oil-well logging.

A natural scale by which to measure inhomogeneities in the static field, B₀, is the amplitude of the RF field B₁. In this disclosure, the term grossly inhomogeneous fields is used to describe those fields in which the inhomogeneities of the static field, ΔB₀, exceed the strength of the RF field, B₁. In this case, the NMR signal spectrum depends mainly on B₁ and the value of the dephasing time of the free induction decay, T*₂, is on the order of the pulse duration. This implies that standard NMR spectroscopy cannot be used encode chemical or spatial information in the signal spectrum. As a consequence, the standard “spectral approach” fails with downhole NMR logging devices that have grossly inhomogeneous fields.

Spin relaxation times, such as the longitudinal relaxation time, T₁, and the transverse relaxation time, T₂, are important for characterization of crude oils. Most NMR logging measurements are currently based on measurements of transverse relaxation times, T₂, because they can be measured very efficiently using a Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence. The CPMG sequence generates a long train of echoes whose amplitudes decay with the time constant T₂. In this case, the echo amplitudes provide the essential information, in particular, the initial amplitude and the decay time. The measurement of the longitudinal relaxation time, T₁, is more time consuming. However, there are circumstances when T₁ measurements may be more desirable than T₂ measurements, particularly when the intrinsic relaxation times are long, for example, greater than one second. In such cases, the intrinsic T₂ sometimes cannot be determined because the measurement becomes dominated by diffusion and motion effects, whereas in contrast, T₁ is not affected by diffusion and is less affected by motion effects.

A large number of T₁ measurement techniques can be found in the literature. The majority of schemes are based on inversion recovery or saturation recovery. This requires measurements with long recovery times (e.g., greater than several times T₁) to determine the equilibrium magnetization, M₀. In samples with long T₁, this results in very lengthy measurement cycles. There are many techniques that attempt to speed up measurements of T₁, such as measuring the approach to steady-state magnetization, progressive saturation measurements (in which a series of steady state signals with different relaxation weightings are measured), speed-optimized fast-inversion recovery (FIR) methods, and many others. However, common to most existing methods is the requirement to take one or several data for recovery times much longer than T₁ in order to obtain the equilibrium signal.

Single-scan measurements are the fastest T₁ measurement schemes. Many of these schemes are a modification of the so-called “triplet method,” in which the recovering longitudinal magnetization is monitored by briefly converting it into transverse magnetization, detecting it, and the restoring it back to longitudinal magnetization. However, in grossly inhomogeneous fields, off-resonance effects prevent complete conversion into transverse magnetization and back. As a result, the measured relaxation time is not a pure T₁ relaxation time, but with a strong admixture of T₂. Another single-scan approach to measure T₁ is based on a standard two-dimensional inversion-recovery sequence, but the second dimension is encoded in the spatial dimension using pulsed field gradients. This allows the second dimension to be encoded simultaneously with the first dimension to reduce the measurement time to that of a one-dimensional experiment. However, this technique is also not easily adapted to grossly inhomogeneous fields.

SUMMARY OF INVENTION

Aspects and embodiments of the invention are directed to methods and pulse sequences for facilitating nuclear magnetic resonance (NMR) measurements in grossly inhomogeneous fields. There are described herein methods and pulse sequences that may be used to accurately measure variables such as transverse relaxation time, longitudinal relaxation time, and diffusion, without the need for data at long recovery time. This may allow faster measurements. In addition, there are described herein methods and pulse sequences that may allow simultaneous encoding of information in both the amplitude and the shape of echoes, so as to allow a single-shot measurement of multiple variables, e.g., both transverse relaxation time (from the decay of echo amplitudes) and longitudinal relaxation time (from the echo shape). CPMG detection may be used to overcome the often limited signal-to-noise ratio in grossly inhomogeneous fields.

According to one embodiment, a method of measuring a longitudinal relaxation time in a sample having an initial magnetization may comprise disturbing the initial magnetization with a first series of RF pulses, after a recovery time period has elapsed, applying a second series of RF pulses to the sample to acquire a first signal comprising at least two echoes, disturbing the initial magnetization differently with a third series of RF pulses, after the recovery time period has elapsed, applying a fourth series of RF pulses to the sample to acquire a second signal comprising at least two echoes, obtaining a difference signal from the first signal and the second signal, and analyzing the difference signal to obtain the longitudinal relaxation time. The measurement may be repeated for a series of recovery times including both short recovery times (i.e., less that the longitudinal relaxation time) and long recovery times (e.g., several times the length of the longitudinal relaxation time).

In one example, the second and fourth series of RF pulses may be substantially identical, and may comprise a Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence. In another example, analyzing the difference signal may comprise fitting a function, such as a single or double exponential function, or a one- or two-dimensional numerical Laplace inversion, to the difference signal. In one example, the first series of RF pulses may comprise a first pair of 90° pulses and the third series of RF pulses may comprise a second pair of 90° pulses, wherein the first and second pairs of 90° RF pulses may have different phase cycling. In another example, the first series of RF pulses may comprise a 180° pulse and the third series of RF pulses may have no corresponding pulse.

Another embodiment may be directed to a method of measuring a longitudinal relaxation time in a sample having an initial magnetization, the method comprising applying a sequence of RF pulses to the sample, the sequence including an encoding portion and a detection portion, acquiring an echo signal using the detection portion of the sequence of RF pulses, decomposing the echo signal into at least two coherence pathway components; and analyzing at least one of the two coherence pathway components to determine the longitudinal relaxation time. In one example, decomposing the echo signal into at least two coherence pathway components may include decomposing the echo signal into a decay component and a recovery component. In another example, analyzing at least one of the two coherence pathway components may include analyzing the decay component. In one example, the encoding portion of the sequence of RF pulses may comprise a pair of 127° pulses separated from one another by a first time period.

According to another embodiment, a method of measuring diffusion in a sample may comprise applying a sequence of RF pulses to the sample, the sequence including an encoding portion and a detection portion, acquiring an echo signal using the detection portion of the sequence of RF pulses, decomposing the echo signal into at least two coherence pathway components, and analyzing the at least two coherence pathway components to determine a diffusion coefficient. In one example, decomposing the echo signal into at least two coherence pathway components may include decomposing the echo signal into a direct echo component and a stimulated echo component. Analyzing the at least two coherence pathway components may then include extracting a first amplitude of the direct echo component, extracting a second amplitude of the stimulated echo component, and determining the diffusion coefficient from a ratio of the first and second amplitudes. In one example, the detection portion of the sequence of RF pulses may comprise a Carr-Purcell-Meiboom-Gill (CPMG) pulse train. In another example, the encoding portion of the sequence of RF pulses may comprise a 90° pulse and a pair of 180° pulses having phases that differ by 90°.

According to another embodiment, a nuclear magnetic resonance measurement device may comprise a transmitter constructed and arranged to generate a sequence of RF pulses and to apply the sequence of RF pulses to a sample, a receiver constructed and arranged to receive an echo signal from the sample, and a processor constructed and arranged to decompose the echo signal into at least two coherence pathway components, and to analyze the at least two coherence pathway components to determine a parameter of the sample. In one example, the sequence of RF pulses may comprise an encoding portion and a detection portion. The detection portion may comprise a Carr-Purcell-Meiboom-Gill (CPMG) pulse train. The encoding portion may comprise, for example, a pair of 127° pulses separated from one another by a first time period, or a 90° pulse and a pair of 180° pulses having phases that differ by 90°. In one example, the processor may be constructed and arranged to decompose the echo signal into a decay component and a recovery component. In one example, the processor may be constructed and arranged to analyze the decay component to obtain a measurement of a longitudinal relaxation time of the sample. In another example, the processor may be constructed and arranged to decompose the echo signal into a direct echo component and a stimulated echo component. The parameter of the sample may be a diffusion coefficient, and the processor may be constructed and arranged to analyze the direct echo component to extract a first amplitude, and to analyze the stimulated echo component to extract a second amplitude of the stimulated echo component; and wherein the processor is configured to determine the diffusion coefficient from a ratio of the first and second amplitudes.

BRIEF DESCRIPTION OF THE DRAWINGS

Various aspects and embodiments of the invention are described below with reference to the accompanying figures. In the drawings, which are not intended to be drawn to scale, each identical or nearly identical component that is illustrated in various figures is represented by a like numeral. For purposes of clarity, not every component may be labeled in every drawing. In the drawings:

FIG. 1 is a diagram of a standard inversion recovery CPMG sequence using nominal 180° and 90° pulses;

FIG. 2A is a diagram of the calculated echo shape of the in-phase signal for the decaying coherence pathway;

FIG. 2B is a diagram of the calculated echo shape of the in-phase signal for the recovering coherence pathway;

FIG. 3 is a diagram of example echo shapes of the in-phase signal for different recovery times between 1 ms and 10 s;

FIG. 4 is a plot showing the covariance matrix diagonal elements for inversion recovery and decay examples;

FIG. 5A is a diagram of a pulse sequence for a first measurement scan according to one embodiment of the invention;

FIG. 5B is a diagram of a pulse sequence for a second scan, that forms a pair with the pulse sequence of FIG. 5A, according to one embodiment of the invention;

FIG. 6A is a diagram of another pulse sequence for a first measurement scan according to another embodiment of the invention;

FIG. 6B is a diagram of a pulse sequence for a second scan, that forms a pair with the pulse sequence of FIG. 6, according to an embodiment of the invention;

FIGS. 7A-H are illustrations of theoretical signal shapes in the time domain (FIGS. 7A, 7C, 7E and 7G) and frequency domain (FIGS. 7B, 7D, 7F and 7H) for a set of pulse sequences;

FIGS. 8A-H are illustrations of example signal shapes using a sample containing tap water (Sample A) in the time domain (FIGS. 8A, 8C, 8E and 8G) and frequency domain (FIGS. 8B, 8D, 8F and 8H) for the same set of pulse sequences as used in FIGS. 7A-H;

FIGS. 9A-D are illustrations of the time dependence of measured spectra for decomposition examples using Sample A;

FIG. 10 is a plot of T₁ fit accuracy as a function of measurement range

FIG. 11 is a plot of normalized measured T₁ values as a function of measurement range;

FIGS. 12A-C are T₁-T₂ maps from examples performed on a composite water sample (Sample B) using a SPARLM2d pulse sequence:

FIGS. 12D-F are T₁-T₂ maps from examples performed on Sample B using an SR pulse sequence;

FIGS. 13A-C are T₁-T₂ maps from examples performed on a composite water and rock sample (Sample C) using a SPARLM2d pulse sequence;

FIGS. 13D-F are T₁-T₂ maps from examples performed on Sample C using an SR pulse sequence;

FIG. 14 is a plot of the calculated spectrum of the longitudinal magnetization after the initial 180° pulse produced by a standard inversion recovery CPMG sequence;

FIG. 15 is a plot of the calculated spectrum of the longitudinal magnetization after the initial inversion pulses produced by a modified inversion recovery CPMG sequence, according to an embodiment of the invention;

FIG. 16 is a timing diagram of a pulse sequence according to an embodiment of the invention;

FIG. 17A is a plot of the spectrum of the decaying coherence pathway for a modified inversion recovery CPMG sequence according to an embodiment of the invention;

FIG. 17B is a plot of the spectrum of the recovering coherence-pathway for the same modified inversion recovery CPMG sequence according to an embodiment of the invention;

FIG. 17C is an illustration of the time domain echo shape for the decaying coherence pathway for the same modified inversion recovery CPMG sequence according to an embodiment of the invention

FIG. 17D is an illustration of the time domain echo shape for the recovering coherence pathway for the same modified inversion recovery CPMG sequence according to an embodiment of the invention;

FIG. 18A is a plot of the in-phase component of example echo shapes with the modified pulse sequence according to an embodiment of the invention.

FIG. 18B, is a plot of the out-of-phase component of example echo shapes with the modified pulse sequence according to an embodiment of the invention;

FIG. 19 is a plot of signal power (represented on the vertical axis) versus a ratio of the relaxation time, T₁, to the recovery time, τ, (represented on the horizontal axis) for different pulse sequences:

FIG. 20 is as plot of extracted amplitudes for the decaying coherence pathway and recovering coherence pathway:

FIG. 21 is a plot of the relationship between the longitudinal relaxation time and the relative size of the amplitudes of the decaying and recovering coherence pathways;

FIG. 22 is a timing diagram of one example of a diffusion editing pulse sequence;

FIG. 23 is a timing diagram of another example of pulse sequence that may be used for diffusion encoding by phase separation, according to an embodiment of the invention;

FIG. 24A is a plot of a calculated echo shape for the direct echo coherence pathway using the pulse sequence of FIG. 22;

FIG. 24B is a plot of a calculated echo shape for the stimulated echo coherence pathway using the pulse sequence of FIG. 22;

FIG. 25A is a plot of a calculated echo shape for the direct echo coherence pathway using the pulse sequence of FIG. 23;

FIG. 25B is a plot of a calculated echo shape for the stimulated echo coherence pathway using the pulse sequence of FIG. 23;

FIGS. 26A-E are plots of example echo shapes for the pulse sequence of FIG. 23 for a series of diffusion coefficient values;

FIGS. 27A-E are plots of extracted amplitudes versus time of in-phase and out-of-phase signal components for the corresponding echo shapes of FIGS. 26A-E;

FIG. 28 is a plot of fitted amplitudes of the in-phase and out-of-phase echo signals versus the diffusion coefficient for the pulse sequence of FIG. 23;

FIG. 29 is a timing diagram of a pulse sequence for diffusion encoding by time separation according to an embodiment of the invention;

FIG. 30A is a plot of a calculated echo shape for the direct echo coherence pathway using the pulse sequence of FIG. 29;

FIG. 30B is a plot of a calculated echo shape for the stimulated echo coherence pathway using the pulse sequence of FIG. 29;

FIGS. 31A-E are plots of example echo shapes for the pulse sequence of FIG. 29 for a series of diffusion coefficient values; and

FIG. 32 is a plot of fitted amplitudes of the in-phase and out-of-phase echo signals versus the diffusion coefficient for the pulse sequence of FIG. 29.

DETAILED DESCRIPTION

Oilfield fluids may contain components that exhibit very long spin relaxation times, particularly, long longitudinal spin relaxation times, T₁. For example, oils with high gas-to-oil ratio (GOR) at high temperatures may present a T₁ on the order of about 10 seconds (s). However, conventional methods for measuring T₁, such as inversion recovery and saturation recovery, require data with recovery times comparable to, or several times larger than, T₁, making it difficult for continuous logging. Aspects and embodiments of the invention, therefore, are directed to methods for obtaining accurate measurements of T₁, particularly in inhomogeneous fields, without the need for data at long recovery times. Techniques according to various aspects of the invention may allow much faster acquisition of well-logging data, particularly for very light oil with high GOR, and/or a significant improvement in the accuracy of T₁ measurements.

According to one embodiment, there are provided pulse sequences that generate multiple echoes and are suitable for logging and other applications in inhomogeneous fields. Information may be extracted from an analysis of the echo amplitudes and/or echo shapes. In one embodiment, sequences are provided, using amplitude encoding, to measure transverse relaxation times, T₂, longitudinal relaxation times, T₁, diffusion coefficients, D, and the corresponding one-dimensional and two-dimensional distribution functions between these quantities, as discussed further below. In another embodiment, information about the above-mentioned properties may also be encoded in the echo shape, which may be particularly useful for measuring quantities such as T₁ and D that are time consuming to measure with conventional approaches. This echo shape encoding maybe combined with echo amplitude encoding, leading to significant decreases in measurement time, and/or increase in information obtained.

The standard Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence comprises an initial nominal 90° pulse that generates transverse magnetization. For multi-dimensional measurements, the initial 90° pulse may be replaced by a preparation pulse sequence that may be varied to change the sensitivity to a quantity of interest. For example, the preparation sequence may be an inversion recovery sequence (180°-τ-90°) to encode T₁ information or a stimulated echo sequence (90°-δ-90°-T_(d)-90°-δ) to encode diffusion information (where T_(d) and δ are pulse spacings. T_(d) often being referred to as the diffusion time, and δ being the gradient encoding time). This initial pulse sequence is then followed by a long series of closely-spaced 180° refocusing pulses. Referring to FIG. 1, there is illustrated one example of a standard inversion recovery CPMG pulse sequence. The inversion recovery sequence, 180°-τ-90°, includes nominal 180° and 90° pulses 100 and 102, respectively, separated by a time, τ. The initial RF pulse 100 modifies the equilibrium magnetization. Next, the second RF pulse 102 is applied after some time, τ, has elapsed in order to monitor the recovery of the magnetization to equilibrium. The initial pulse sequence is followed by the series of refocusing pulses 104. The refocusing pulses are separated from one another by a time, t_(E). In CPMG-like sequences, the repeated refocusing may serve two purposes. First, many echoes may be averaged to obtain the echo shape S(t). This may help to overcome the intrinsically small signal-to-noise ratio in grossly inhomogeneous fields. Second, the time dependence of the echo amplitudes allows measurement of the magnetization spin relaxation times, as discussed further below.

In grossly inhomogeneous fields, the spectrum of echoes from a CPMG-like sequence quickly approaches an asymptotic form and may be given by:

S(ω₀)=({right arrow over (M)} _(A) ·{circumflex over (n)}){circumflex over (n)}  (1)

For the standard CPMG sequence, on resonance, S(ω=0)=M₀, the thermal equilibrium magnetization. The magnetization of the k-th echo may be given by:

{right arrow over (M)} _(k)=({right arrow over (M)} _(A) ·{circumflex over (n)}){circumflex over (n)} exp(−kt _(E) /T _(2,eff))   (2)

where: {right arrow over (M)}_(A) is the magnetization after the initial preparation period;

-   -   t_(E) is the echo spacing; and     -   T_(2,eff) is the transverse spin relaxation time.

According to one embodiment, to calculate M_(A) and its dependence on parameters such as diffusion or T₁, the magnetization evolution may be divided into different coherence pathways. Each coherence pathway may have a distinct dependence on parameters of interest, such as relaxation times or diffusion coefficients, and may generate a signal of known echo shape. The measured echo shape may then be decomposed into contributions from the individual coherence pathways and the relative amplitudes may be directly related to the parameters of interest, as discussed further below.

For any coherence pathway, i, the echo signal in grossly inhomogeneous fields may be factorized into a spectral portion s_(i)(ω₀)) and an amplitude a_(i)(D, T₁, . . . ). The spectral portion s_(i)(ω₀) may depend only on the applied RF pulses. On the other hand, the effects of diffusion, relaxation, flow or pulsed field gradients perpendicular to the static field gradient may be uniform across the spectrum and may be described by the amplitude a_(i). This amplitude may depend on the duration of the time intervals between the RF pulses. The asymptotic spectrum of echoes from equation (2) may be written as a sum over the different coherence pathways:

$\begin{matrix} {{S\left( \omega_{0} \right)} = {{\left( {{\overset{\rightarrow}{M}}_{A} \cdot \hat{n}} \right)\hat{n}} = {\sum\limits_{i = 1}^{N}{{s_{i}\left( \omega_{0} \right)}{a_{i}\left( {D_{i},T_{1},\ldots} \right)}}}}} & (3) \end{matrix}$

Since different coherence pathways have, in general, different sensitivities to diffusion or T₁, the spectrum S(ω₀), or echo shape in the time domain, of the echo signal may depend on these parameters. Thus, information may be encoded not only in the amplitude of the echo signal, but also in the echo shape. This allows a two-dimensional measurement from a single echo pulse sequence.

In grossly inhomogeneous fields, the signal-to-noise ratio of a single echo acquisition may be insufficient to allow a detailed analysis of its shape. This may be overcome by using pulse sequences that comprise an initial preparation or encoding sequence followed by a long train of refocusing pulses (e.g., a CPMG sequence). The CPMG-like sequence may serve two distinct roles. First, the decay of the echo amplitudes may give a measurement of T₂. Second, it may greatly increase the precision of the echo shape measurement because, after the first few pulses, the echo shapes may not change and therefore, a large number of echoes may be averaged to obtain a robust shape with good signal-to-noise ratio. Further, in one embodiment, to optimize the “single-shot” measurement approach according to aspects of the invention, preparation sequences may be chosen with significant contributions from different coherence pathways that exhibit greatly different sensitivities to a parameter of interest (e.g., T₁) and that also have echo shapes that are as orthogonal to each other as possible. Examples of such preparation sequences are discussed below.

According to one embodiment, preparation sequences may be selected such that at least two coherence pathways contribute to the echo signal. This may be demonstrated with reference to the inversion recovery sequence combined with CPMG detection shown in FIG. 1. After the recovery time, τ, at the time of the 90° pulse 102, the magnetization may have components from two different coherence pathways referred to as the “decay” component (from the decaying coherence pathway) and the “recovery” component (from the recovering coherence pathway). If the initial magnetization of a sample is M₀, then the spectrum of echoes after the first few refocusing pulses may be obtained from equation (3) and may be given by:

S(ω₀)=s _(d)(ω₀) exp{−τ/T ₁ }+s _(r)(ω₀)(1−exp {−τ/T ₁})   (4)

The first term of equation (4) is the contribution from the decaying coherence pathway and the second term is the contribution from the recovering coherence pathway. The spectra of the individual coherence pathways may be given by:

s _(d)(ω₀)=M ₀ Im((Λ_(+1,−1) ⁽³⁾Λ_(−1,0) ⁽²⁾Λ_(0,0) ⁽¹⁾)e ^(−iω) ⁰ ^(t) ¹⁸⁰ ^(/π) n _(y) ²)   (5)

s _(r)(ω₀)=M ₀ Im((Λ_(+1,−1) ⁽³⁾Λ_(−1,0) ⁽²⁾)e ^(−iω) ⁰ ^(t) ¹⁸⁰ ^(/π) n _(y) ²)   (6)

where: Λ_(i,j) ^((k)) are transition probabilities for the k-th pulse between three different magnetization components (M₊₁, M⁻¹, M₀);

-   -   t₁₈₀ is the time at the first of the series of 180° pulses 104         (see FIG. 1);     -   n_(y) is the y-component of the effective rotation axis for the         refocusing cycle; and     -   Im indicates that the imaginary component of the expression in         brackets is important.         In one example, to maximize the signal, the pulse spacing         between the 90° pulse 102 and the first 180° refocusing pulse         may be reduced from half the echo spacing by t₁₈₀/π, as shown in         FIG. 1. This reduction is reflected in equations (5) and (6) by         the extra phase shifts of exp(−ω₀t₁₈₀/π).

Based on equations (5) and (6), echo shapes due to the decay and recovery components may be calculated. For an extended sample along the field gradient direction, the echo shapes in the time domain are the Fourier transforms of equations (5) and (6). It is to be appreciated that after the first few echoes, all subsequent echoes may have substantially the same shape in the time domain and it is thus still possible to average many echoes to achieve good statistics. In the time domain, the overall echo shape may be given by a weighted sum of the characteristic echo shapes from the decaying, coherence pathway and the recovering coherence pathway:

S(t)=s _(r)(t)(1−exp(−τ/T ₁))+s _(d)(t)exp(−τ/T ₁)   (7)

In equation (7), the first term is referred to herein as the recovery component and the second term is referred to as the decay component.

Referring to FIGS. 2A and 2B, there are illustrated calculated echo shapes for the decaying (FIG. 2A) and recovering (FIG. 2B) coherence pathway for the pulse sequence of FIG. 1. Both echo shapes have only contributions in phase with the 180° refocusing pulses. They differ significantly from one another so that the measured echo shape can be decomposed into the two shapes, s_(r)(t) (the recovery component) and s_(d)(t) (the decay component), as discussed below.

An example was performed to confirm the calculated results discussed above and illustrated in FIGS. 2A and 2B. In this example, a sample of doped water having a longitudinal relaxation time T₁=113.5 milliseconds (ms) was placed in the fringe field outside a superconducting magnet. The RF frequency was set to 1.764 Megahertz (MHz) and the static gradient was 132 millitesla per meter (mT/m). For most measurements, approximately 1000 to 2000 echoes were acquired in the CPMG train. The diameter of the sample was 38 millimeters (mm), which was much larger than the typical slice thickness (approximately 5 mm), such that to first order, the finite sample size may be neglected. The echo spacing, t_(E), was chosen to be 384 μs and t₁₈₀=24 μs. In one example, the recovery time, τ, was varied logarithmically between 1 millisecond (ms) and 10 seconds (s). For the measurement at each recovery time, the signal from the 10^(th) to the 110^(th) echo was averaged to obtain the echo shapes illustrated in FIG. 3. For short recovery times, τ<<T₁, the measured echo shapes coincide with the shape of s_(d)(t) (shown in FIG. 2A), whereas in the other limit, τ>>T₁, the echo shapes approach s_(r)(t) (shown in FIG. 2B). This example therefore confirms the sensitivity of the echo shape to T₁. In addition, the example suggests the usefulness of the decay component in determining T₁ for short recovery times at which the composite echo shape tends to approach that of the decay component alone.

For a standard inversion recovery CPGM sequence, such as shown in FIG. 1, and for simplicity setting s_(r)(t)=M₀, equation (7) can be rewritten:

S(τ)∝M₀(1−α exp(−τ/T₁))   (8)

where α is determined from M₀ and s_(d)(t). When the exponent is much less than unity (i.e., τ/T₁<<1), corresponding to a recovery time much smaller than the longitudinal relaxation time, then:

S 96 )≈M ₀(1−α)+(αM ₀ /T ₁)τ  (9)

At short recovery time, τ, the signal shows a linear τ dependence with a slope of αM₀/T₁. This demonstrates that from very short recovery time data, it is impossible to determine the values of M₀ and T₁ independently. On the other hand, if one acquires the decay signal, rather than a recovery signal, that is, taking the decay component alone from equation (7) and, for simplicity's sake, setting s_(d)(t)=M₀, the echo signal as a function of recovery time is given by:

S(τ)∝M₀ exp(−τ/T₁)   (10)

It can be seen that the two variables, M₀ and τ, are more independent than they are in equation (8). For example, the slope of the logarithm of S (log S) versus τ will determine T₁ directly. Therefore, it may be desirable to determine T₁ from the decay component to avoid the problems of conventional inversion recovery, particularly at short recovery times. Accordingly, aspects and embodiments of the invention are directed to schemes to isolate the two signal components in order to obtain T₁ reliably from data at short τ only, as discussed below. Although there have been some efforts in the prior art to demonstrate the use of the decay component for T₁ measurement, these experiments have been performed in uniform magnetic fields. In contrast, schemes according to various embodiments of the invention are applicable to NMR measurements in grossly inhomogeneous fields, such as in a fringe field as exists in NMR logging tools, as discussed below.

From the above analysis of the short time behavior of the inversion recovery measurement method and the decay measurement method, it is clear that, within the short time limit, the conventional inversion recovery technique cannot determine T₁ independent of the amplitude of the signal. By analyzing the error propagation of the two methods, advantages of using the decay component may be further demonstrated. Referring again to equation (8), there are three parameters present in this fitting, namely, M₀, α, and T₁. By differentiating the right hand side of equation (8), the variance of the signal S may be obtained:

$\begin{matrix} {{\delta \; {S(\tau)}^{2}} = {{\delta \; {M_{0}^{2}\left( \frac{\partial S}{\partial M_{0}} \right)}^{2}} + {\delta \; {T_{1}^{2}\left( \frac{\partial S}{\partial T_{1}} \right)}^{2}} + {2\; \delta \; M_{0}\delta \; {T_{1}\left( {\frac{\partial S}{\partial M_{0}}\frac{\partial S}{\partial T_{1}}} \right)}} + \ldots}} & (11) \end{matrix}$

where: δM₀ is the variance of the parameter M₀;

-   -   δT₁ is the variance of the parameter T₁; and     -   δM₀δT₁ is the covariance of the two parameters         Formally, this relationship can be obtained for a function         ƒ(x,p) with variable x and parameters p by defining the Jacobian         matrix:

$\begin{matrix} {J \equiv \begin{pmatrix} {f_{p\; 1}^{\prime}\left( {x_{1},p} \right)} & {f_{p\; 2}^{\prime}\left( {x_{1},p} \right)} & \cdots \\ {f_{p\; 1}^{\prime}\left( {x_{2},p} \right)} & {f_{p\; 2}^{\prime}\left( {x_{2},p} \right)} & \cdots \\ \cdots & \cdots & \; \\ {f_{p\; 1}^{\prime}\left( {x_{n},p} \right)} & {f_{p\; 2}^{\prime}\left( {x_{n},p} \right)} & \cdots \end{pmatrix}} & (12) \end{matrix}$

where:

-   -   x_(i) is the i^(th) variable (such as τ);     -   p_(i) is the i^(th) parameter (such as T₁); and     -   f′_(pi) is the derivative of the function ƒ with respect to the         i^(th) parameter:     -   (ƒ′_(pi)≡∂ƒ/∂p_(i)).         The covariance matrix may then be obtained through the formula         (J^(T)J)⁻¹, where the exponent −1 indicates matrix inversion.         The diagonal elements of the covariance matrix may be useful in         determining the ratio of the variance of the fitting parameters,         for example, δT₁ ², to that of the data, δS². This ratio may         serve as a quality index for the measurements.

In one example, for both the inversion recovery method and the decay method (i.e. using only the decay component described by equation (10)), the covariance matrix was obtained for an example with 30 values of r over a range of τ from 0.1 T₁ to τ_(max). The diagonal elements from the covariance matrix are plotted in FIG. 4 as a function of τ_(max), with τ_(max) ranging from 0.5 T₁ to 3 T₁. Referring to FIG. 4, line 106 represents M₀ for the decay method example and line 108 represents T₁ for the decay method example. Line 110 represents M₀ for the inversion method recovery example and line 112 represents T₁ for the inversion recovery method example. For both examples, M₀ and T₁ were both assumed to be unity (M₀=1 and T₁=1). Line 114 represents α for the inversion recovery example, which was assumed to be equal to 2. As discussed above, when the recovery time τ_(max) is long (close to 3 T₁), the entire T₁ relaxation may be well observed in both the inversion recovery and the decay examples. Therefore, it is not surprising that both methods obtain similar and small variance for all the parameters. However, as τ_(max) is reduced, the variance for the inversion recovery method example increases dramatically, indicating larger errors in the estimate of the parameters (M₀ and T₁). By contrast, as can be seen in FIG. 4, the variance of these parameters in the decay method example (lines 106 and 108) remains relatively small and shows only a modest increase as τ_(max) approaches 0.5 T₁. These examples show that the inversion recovery method yields a variance about two orders of magnitude greater than does the decay method for the same parameters, providing mathematical evidence of an advantage of the decay method.

According to one embodiment, there is developed a class of preparation pulse sequences that may facilitate separating the recovery and decay components for an improved T₁ measurement. In one embodiment, a preparation pulse sequence may be constructed to acquire two signals by implementing the following steps. In a first scan, the equilibrium magnetization may be disturbed by a series of RF pulses, a time, τ, may then be allowed to elapse, after which a second series of RF pulses may be applied to acquire the first signal. In a second scan, the equilibrium magnetization may be disturbed differently from the first scan, again a time, τ, may be allowed to elapse, and then another series of RF pulses may be applied to acquire the second signal. Once the two signals are acquired, a difference signal (i.e., obtained by subtracting the first signal from the second signal, or vice versa) may be the decay component, and a summation of the two signals may provide the recovery component.

According to one embodiment, a class of pulse sequences that may be used to achieve separation of the decay component according to the above steps is referred to herein as Separation of Preexisting and Longitudinal Magnetization (SPARLM). It is to be appreciated that this acronym, as used herein, refers not only to the example pulse sequences described herein, but also to all analogous pulse sequences. In one example. the SPARLM sequence may comprise two 90° pulses with an (x,x) phase in one scan and another phase, (x,−x) in a second scan. The two 90° degree pulses may be separated by a short delay that may vary from one example to another and may be selected by a person performing the measurements. In one example, the signals in the two scans may be given by:

S ₁ =M ₀[1−exp {−τ/T₁ }]−M ₀ exp {−τ/T ₁}

S ₂ =M ₀[1−exp {−τ/T₁ }]+M ₀ exp {−τ/T ₁}  (13)

S₁ is the signal from the first scan and S₂ is the signal from the second scan. The difference of the signals from the two scans may produce the decay component (as can be seen with reference to equation (10)), and their addition may produce the recovery component. This process of subtraction or addition is referred to herein as decomposition because the effect is to divide the signal into the two-components.

Referring to FIGS. 5A and 5B, there is illustrated one example of SPARLM pulse sequences. The narrower and wider boxes represent 90° and 180° pulses, respectively. FIG. 5A illustrates the signal in the first scan with the two 90° pulses 116 a and 116 b having phases (x,x). FIG. 5B illustrates the signal in the second scan with the pair of 90° pulses 116 a,b having phases (x,−x). Both signals may also include a 90° presaturation pulse 118. This extra 90° pulse may be inserted at the beginning of the sequence to ensure equivalent preparation for all measured transients. The signal recovery during the τ₀ period determines the magnetization before the pair of 90° pulses 116 a,b. It is to be appreciated that the presaturation pulse 118 may be redundant because the long CPMG detection pulse train may establish a steady-state longitudinal magnetization with no memory of previous pulse sequences that may have been applied to the sample. The presaturation pulse 118 may therefore not be required, but may be included to serve as a quality control measure.

Referring to FIGS. 6A and 6B, there is illustrated an example of a variation on the SPARLM pulse sequences of FIGS. 5A and 5B. In this example, the pair of 90° inversion pulses 116 a,b may be merged, producing a 180° pulse 120 in the first scan (FIG. 6A) and no pulse in the second scan (FIG. 6B). This sequence may accomplish the same decay-recovery separation described above, but with a different (and simpler) spatial inversion profile. In this example, the sequences for the two scans resemble a “fast inversion recovery” (FIR) sequence (FIG. 6B) and a saturation recovery (SR) sequence (FIG. 6A). Therefore, this variation on the SPARLM sequence is referred to herein as FIR/SR.

For a pure on-resonance signal, it may be possible to adjust the excitation to achieve a uniform 90° pulse for the entire sample volume. However, when the sample is in a strong constant field gradient, all RF pulses are slice-selective and off-resonance effects may need to be accounted for. For example, a nominal 90° pulse may exhibit a nutation angle that is dependent on position. Therefore, to account for such effects, the signal equation for SPARLM may be modified:

S ₁ =M ₀[1−exp {τ/T ₁ }]+m ₁ exp {−τ/T ₁}

S ₁ =M ₀[1−exp {τ/T ₁ }]+m ₂ exp {−τ/T ₁}  (14)

where m₁ and m₂ may be different from each other and from M₀. However, even with this modification, the difference of S₁ and S₂ may still produce a pure decay signal:

S _(d)=(m ₁ −m ₂)exp {−τ/T ₁}  (15)

Thus, the SPARLM pulse sequences may be compatible with grossly inhomogeneous field examples, as may occur in NMR well-logging tools.

According to another embodiment, m₁ and m₂ may be frequency dependent. This frequency dependence, and thus the spectrum of the resulting echoes may be calculated. The results are illustrated in FIGS. 7A-H. In each figure, the solid line represents the real part and the dashed line represents the imaginary part. FIGS. 7A, 7C, 7E and 7G illustrate theoretical calculations of the signal shapes in the time domain, and FIGS. 7B, 7D, 7F and 7H illustrate corresponding theoretical calculations of the signal shapes (spectra) in the frequency domain. These theoretical signal shapes are obtained from a discrete isochromat simulation of a rectangular object whose size exceeds the slice width of the RF pulses. FIGS. 7A and 7B illustrate the calculated signal shape of a saturation recovery (SR) pulse sequence (see FIG. 6A) in the time and frequency domains, respectively. FIGS. 7C and 7D illustrate the signal shapes for a fast inversion recovery (FIR) sequence (see FIG. 6B) in the time and frequency domains, respectively. FIGS. 7E and 7F illustrate the calculated signal shapes for the decay component of FIR/SR decomposition in the time and frequency domains, respectively. Lastly, FIGS. 7G and 7H illustrate the calculated signal shapes for the decay component of SPARLM decomposition in the time and frequency domains, respectively.

To verify these theoretical calculated echo shapes, an NMR example was performed on a sample in the fringe field of a Nalorac 2 T superconducting magnet (available from Nalorac Cryogenics) along the axis of the magnet using a TecMag Apollo spectrometer (available from TecMag, Houston, Tex.). The Larmor frequency was 1.7 MHz and the static field gradient was 13 G/cm. RF pulse widths of t_(π/2)=15.6 μs (for the 90° pulses) and t_(π)=31.2 μs (for the 180° pulses) were used. In one example, the sample was a 37 mm ID cylinder and contained tap water, which has a T₁ approximately equal to 2.5 s. This sample is referred to herein as Sample A. The field gradient was oriented along a diameter of the cylinder so that the resonant slice (thickness approximately 5 mm) was a slab oriented along the length of the cylinder. Referring to FIGS. 8A-H there are illustrated measured echo shapes for obtained for Sample A using the same signals for which the theoretical shapes were calculated and illustrated in FIGS. 7A-H. Again, the solid lines represent the real part and the dashed lines represent the imaginary part. FIGS. 8A, 8C, 8E and 8G illustrate the measured signal shapes in the time domain and FIGS. 8B, 8D, 8F and 8H illustrate the corresponding measured signal shapes in the frequency domain. Thus, FIGS. 8A and 8B illustrate the measured signal shape of the saturation recovery (SR) pulse sequence (see FIG. 6A) in the time and frequency domains, respectively. FIGS. 8C and 8D illustrate the measured signal shapes for the fast inversion recovery (FIR) sequence (see FIG. 6B) in the time and frequency domains, respectively. FIGS. 8E and 8F illustrate the measured signal shapes for the decay component of FIR/SR decomposition in the time and frequency domains, respectively. Lastly, FIGS. 8G and 8H illustrate the measured signal shapes for the decay component of SPARLM decomposition in the time and frequency domains, respectively. As can be seen from a comparison of FIGS. 7A-H and FIGS. 8A-H, there is a very good agreement between the example and theory. This indicates that the coherence pathway formalism may provide a complete picture of the spin dynamics both on- and off-resonance.

Referring to FIGS. 9A-D, there is illustrated the time dependence of the example spectra taken for Sample A for the set of pulse sequences discussed above. Frequency domain data are shown as a function of longitudinal magnetization evolution time, τ₁, with logarithmic spacing between spectra. All data in this example were acquired for a fringe field frequency of 1.7 MHz and a static field gradient of 13 G/cm. FIGS. 9A, 9B and 9C illustrate the results from a FIR/SR decomposition example. Specifically. FIG. 9A illustrates FIR spectra and FIG. 9B illustrates SR spectra. FIG. 9C illustrates FIR/SR decay component spectra. As can be seen from FIGS. 9A and 9C, the decomposition process may isolate the region of the spectrum that has been perturbed by the inversion. In other words, the decomposition separates the “hole” from the background spectrum. This hole then decays via T₁ relaxation without substantial change in shape, as shown in FIG. 9C. FIG. 9D illustrates the decay component of the SPARLM example. Qualitatively, this shape shares the time-independent quality of the decay component of the FIR/SR sequence, but shows some additional structure from the multiple pulses involved in the SPARLM inversion sequence. This illustrates the possibility of more complicated inversion profiles while retaining the speed of the T₁ decomposition technique.

Having discussed several signal shapes for T₁ decomposition examples, there are now presented some examples that illustrate the potential speed of T₁ measurement that these signal shapes may provide. As mentioned above, one advantage of a decomposition method according to aspects of the invention is the measurement of a simple decay curve which can be obtained with measurement times, τ₁, significantly below the 3-5 T₁ range required for a recovery example.

In one example, the SPARLM method was tested with Sample A (a water sample) which exhibits a single relaxation time, T₁, of approximately 2.5 s. A set of SPARLM examples was performed on Sample A placed in a field gradient of 13 G/cm. Each example included four measurement points, τ₁, over a range of measurement time. Referring to FIG. 10, there is illustrated a single exponential fit M=M₀exp(−τ₁/T₁) for T₁ measurement accuracy as a function of measurement range to the complete data set. The solid line 122 represents the fit and a solid dots 124 represent measured data points. Referring to FIG. 11, there are illustrated measured T₁ values (normalized to an expected T₁ value of 2.5 s) as a function of the upper limit of each four-point data range, τ_(1max). Accurate results (within 5%) were observed for measurement times as short as approximately 0.05 T₁.

Two-dimensional relaxation examples were performed using both SPARLM and conventional saturation-recovery sequences to demonstrate an advantage of using SPARLM in measuring the long T₁ components in a mixture. In a first example, the sample used referred to herein as Sample B, contained two separate water compartments with different T₁ values of 2.5 s and 0.1 s. In a second example, the sample used, referred to herein as Sample C, included a water saturated Berea sandstone rock in one compartment and tap water in another compartment. The tap water was diluted with unprotonated deuterium oxide (D₂O) to make its contribution to the magnetization comparable to that of the water-saturated rock. The Berea rock possesses a distribution of relaxation times over the approximate range: 10 ms<T₁<1 s, owing to its distribution of pore sizes and the dominant surface relaxation mechanism.

For Sample B, SPARLM2d examples (i.e., using a two-dimensional (2d) SPARLM sequence) were performed with a series of τ₁ delays detected by 2000 echoes (using CPMG detection). SR examples (i.e., using the SR sequence) were also performed on the Sample B with the same τ₁ list and echo number. Two-dimensional Laplace transforms using a fast Laplace inversion (FLI) algorithm were performed to obtain two-dimensional T₁-T₂ spectra in three cases for the two types of examples (SPARLM2d and SR) separately. Three different analysis cases were used, having different τ₁ lists with a maximum τ₁ (τ_(1max)) of 1, 0.0251 and 0.063 seconds, respectively. As stated above, the long T₁ component in Sample B is 2.5 s and thus, the three cases correspond to τ₁/T₁ values of 0.4, 0.1 and 0.0025. In order to focus upon measurement rather than prepolarization effects, τ₀ was set conservatively to 3 s. These examples illustrate the ability of the two types of pulse sequences (SPARLM2d and SR) to detect the long T₁ component with progressively shortened recovery time τ₁.

Referring to FIGS. 12A-F, there are illustrated the resulting two-dimensional T₁-T₂ spectra from these examples and analysis for Sample B. FIGS. 12A-C illustrate the results from the SPARLM2d examples and FIGS. 12D-F illustrate the results from the SR examples. In each figure, three contours of the distribution have been drawn if present in the data. The values of these contours are the same in each of FIGS. 12A-F and were chosen to be 10%, 50% and 90% of the maximum signal in the SPARLM2d distribution for τ_(1max)=1 s. FIG. 12A illustrates the distribution from the SPARLM2d example with τ_(1max)=1 s. FIG. 12B illustrates the distribution from the SPARLM2d example with τ_(1max)=0.251 s. FIG. 12C illustrates the distribution from the SPARLM2d example with τ_(1max)=0.063 s. FIG. 12D illustrates the distribution from the SR example with τ_(1max)=1 s. FIG. 12E illustrates the distribution from the SR example with τ_(1max)=0.251 s, and FIG. 12F illustrates the distribution from the SR example with τ_(1max)=0.063 s. In each case, T₁ is represented on the vertical axis and T₂ is represented on the horizontal axis, both in units of seconds. In each figure, the intensity of the longer T₁ peak includes, at most, the 10% contour, and in some cases, none at all (FIGS. 12E,F).

Referring to FIG. 12A, two peaks 126 a, 128 a centered at T₁=0.1 s and T₁=3 s, respectively, are clearly visible. Similar peaks 126 d and 128 d, also approximately centered at T₁=0.1 s and T₁=3 s are visible in FIG. 12D. A comparison of FIG. 12A and FIG. 12D shows that the spectra are fairly similar in the appearance of both peaks, indicating that both examples are capable of measuring the longer T₁ at a recovery time τ_(1max)=1 s. The relative broadening of the long T₁ peaks 128 a,d is due to insufficient recovery time, τ₁, range compared to the long T₁ component. As noted above, in the examples corresponding to FIGS. 12A and 12D, τ_(1max) is approximately equal to 0.4 times the long T₁ component. These results illustrates that, at the longest recovery time, both sequences work well and the SPARLM results are slightly better (less broadening of the peaks indicates a more accurate measurement). FIGS. 12B and 12E were obtained with τ_(1max)=0.251 s, which is only one tenth of the long T₁ component. Comparing FIGS. 12A and 12B, it can be seen that the long T₁ peak 128 b is further broadened compared to the peak 128 a in FIG. 12A. However, it retains similar signal amplitude and location along both the T₁ and T₂ axes. In contrast, the spectrum from the SR example, illustrated in FIG. 12E, shows a long T₁ peak 128 e with reduced amplitude and shifted position (T_(1,apparent)=1 s). The shorter T₁ peak 126 e is also significantly affected and shows a large broadening along the T₁ (vertical) dimension. These results indicate that, at the shorter recovery time, the SPARLM sequence performs significantly better and may yield a more accurate T₁ measurement. FIGS. 12C and 12F, obtained with τ_(1max)=0.063 s, only 0.0025 times the long T₁ component, demonstrate an even more dramatic comparison between the two sequences. In the SR example (FIG. 12F), the long T₁ peak is completely absent and even the short T₁ peak 126 f is significantly affected. In contrast, the SPARLM example, data (FIG. 12C) still shows a clear separation between the two T₁ peaks 126 c, 128 c without drastic change in either their location or amplitude. Thus, these examples demonstrate that the SPARLM sequence decomposition technique discussed above is useful for measuring long relaxation times without requiring long measurement time.

Examples using Sample C were also performed analogously to those described above using Sample B. In the Sample C series of examples, the measurement time limits employed were τ_(1max)=2.5 s, 0.631 s, and 0.158 s, with a prepolarization time, τ₀, of one second. FIGS. 13A-13F illustrate the resulting relaxation spectra from these examples. The contours displayed are the same for each figure and (if present) are at 10%, 50% and 90% of the maximum signal in the SPARLM2d distribution for τ_(1max)=2.5 s. FIGS. 13A, 13B and 13C illustrate the distributions obtained from the SPARLM2d examples and FIGS. 13D, 13E and 13F illustrate the distributions obtained from the saturation recovery examples. As was the case for the Sample B data, the SPARLM2d spectra are more robust than are the saturation recovery spectra as a function of shortened measurement time τ_(1max). In all the SPARLM2d spectra, the signal 130 from the water confined in the Berea sandstone can be qualitatively distinguished from the free tap water signal 132 at the long relaxation time region of the spectrum. It is noted that the signal 130 shows a distribution of relaxation times due to surface relaxation from a range of pore sizes in the rock. In contrast, in the saturation recovery examples, the two signals are completely indistinguishable for τ_(1max)=0.158 s=0.5 T_(1,water) (FIG. 13F) while some separation between them still exists in the data from the SPARLM2d sequence (FIG. 13C). The SPARLM examples show less distortion with shortening of the measurement time, τ_(1max), and retains sensitivity to the longest T₁ component (i.e., from the tap water). These results show the applicability of SPARLM2d to more realistic samples containing a range of relaxation times, as is ubiquitous among water and oil-saturated rocks.

As discussed above, according to some aspects of the invention, improved T₁ measurements may be achieved through the decomposition of the measured echo shape into two components, the decay component and the recovery component. According to one embodiment, for any value of recovery time, the example echo shapes may be decomposed into a weighted sum of s_(r)(t) and s_(d)(t):

S _(y)(t)=a _(r) s _(r)(t)+a _(d) s _(d)(t)   (16)

Given the echo shapes s_(r)(t) and s_(d)(t), the amplitudes a_(r) and a_(d) can be extracted from a measured echo shape S(t) by the following equations:

$\begin{matrix} {a_{d} = \frac{{\left( {s_{d} \cdot S} \right)\left( {s_{r} \cdot s_{r}} \right)} - {\left( {s_{d} \cdot s_{r}} \right)\left( {s_{r} \cdot S} \right)}}{{\left( {s_{d} \cdot s_{d}} \right)\left( {s_{r} \cdot s_{r}} \right)} - {\left( {s_{d} \cdot s_{r}} \right)\left( {s_{r} \cdot s_{d}} \right)}}} & (17) \\ {a_{r} = \frac{{\left( {s_{r} \cdot S} \right)\left( {s_{d} \cdot s_{d}} \right)} - {\left( {s_{r} \cdot s_{1}} \right)\left( {s_{d} \cdot S} \right)}}{\left( {{\left( {s_{d} \cdot s_{d}} \right)\left( {s_{r} \cdot s_{r}} \right)} - {\left( {s_{d} \cdot s_{r}} \right)\left( {s_{r} \cdot s_{d}} \right)}} \right)}} & (18) \end{matrix}$

where: (s_(d)·S)=∫dts*_(d)(t)S(t) and other similar terms are integrals over the acquisition window.

The relative sizes of these amplitudes may be directly related to the longitudinal relaxation time by the equation:

$\begin{matrix} {\frac{a_{d}}{a_{d} + a_{r}} = {\exp \left\{ {{- \tau}/T_{1}} \right\}}} & (19) \end{matrix}$

Thus, by decomposing the signal into the decay and recovery components, as discussed above, and solving for the amplitudes, T₁ may be directly calculated from a measured echo shape.

In one embodiment, it may be preferable that the decomposition of the echo shape can be performed even with data of limited signal-to-noise ratio (as is often the case with data collected in grossly inhomogeneous fields). It may be particularly preferable if the components are orthogonal (or nearly orthogonal) to each other. In other words, the magnitude of the cross term (s_(d)·s_(r)) may be minimized, while keeping (s_(d)·s_(d)) and (s_(r)·s_(d)) as large as possible. For the standard CPMG sequence shown in FIG. 1, the two echo shapes (see FIGS. 2A and 2B) have a strong overlap and are far from orthogonal to each other. Quantitatively, the normalized cross term in this case is:

(s _(d) ·s _(r))/√{square root over ((s _(d) ·s _(d))(s _(r) ·s _(r)))}{square root over ((s _(d) ·s _(d))(s _(r) ·s _(r)))}=−0.625   (20)

This corresponds to an angle of approximately 129 degrees.

The general approach to the decomposition may not depend on the details of the pulses used. Therefore, it may be possible to improve the performance of the sequence of FIG. 1 by using more sophisticated pulses, such as composite pulses, frequency- or amplitude-modulated pulses, or the pulse sequences discussed above. In one embodiment, a pulse sequence may be designed such that the two components (i.e., the decay component and the recovery component) are substantially orthogonal. As discussed further below, in one embodiment, this may be achieved by designing the pulse sequences such that the different coherence pathways form echoes at different times. In another embodiment, orthogonality may be created by generating signals that are out of phase with one another. Although, in general, this may not be possible in the frequency domain because the refocusing pulses refocus only a single component in grossly inhomogeneous fields, it is possible in the time domain. For example, a decaying signal may be generated that is substantially anti-symmetric with respect to the offset frequency, ω₀, and a recovering signal that is symmetric with respect to ω₀. In this case, the two signals will form out of phase in the time domain. This may be achieved by using an inversion pulse that generates a longitudinal magnetization anti-symmetric with respect to the offset frequency, ω₀.

The standard 180° inversion pulse 100 shown in FIG. 1 generates a symmetric profile in the longitudinal magnetization, as shown in FIG. 14. As can be seen in FIG. 14, the magnetization 134 is clearly symmetric about ω₀/ω₁. According to one embodiment, a solution to create an anti-symmetric spectrum may include using a composite pulse sequence of the form:

127°_(x)−t₉₀−127°_(±y)   (21)

In this embodiment, two nominal 127° pulses (of duration √{square root over (2)} times longer than the nominal 90° pulse, t₉₀) are separated by a time, t₉₀, of free precession. This composite pulse may generate a large anti-symmetric component within ±2ω₁ (where ω₁ is the RF nutation frequency) of the offset frequency. In one example, at ω₀=±ω₁, the resulting longitudinal magnetization is ±M₀. In other words, in this example, the maximum magnitude is anti-symmetric with respect to frequency. FIG. 15 shows the longitudinal magnetization after the composite inversion pulse described by expression (21). It can be seen that this magnetization 136 is not symmetric about ω₀/ω₁. It is to be appreciated that other techniques may also be used to generate anti-symmetric spectrums and that the invention is not limited to the use of the example described above. For example, another means by which to create an anti-symmetric spectrum includes use of the half-adiabatic-fast-passage.

According to one embodiment, to refocus the magnetization produced by the preparation pulse sequence of expression (21), an off-resonance CPMG sequence may be used having the form:

127_(s)−(t_(E)/2+τ_(α))−(127_(x)127_(−x)−t_(E))^(N)   (22)

where: τ_(a)=−t₉₀.

With this sequence, the magnetization at ±ω₁ may be completely refocused. Thus, combining expressions (21) and (22), a modified inversion recovery CPMG sequence according to one embodiment of the invention may have the form:

(127°_(x)−t₉₀−127°_(±y))−τ−(127_(x)−(t_(E)/2+τ_(a))−(127_(x)127_(−x)−t_(E))^(N))   (23)

An example of such a modified inversion recovery CPMG sequence is illustrated in FIG. 16. The sequence includes a composite inversion pulse 138, having the form described in expression (21), followed by a series 140 of refocusing pulses that begin after the recovery time, τ, has elapsed, t_(E) is the echo spacing, and t₉₀ and t₁₈₀ are the durations of nominal 90° and 180° pulses, respectively. With the phase of pulses and timing shown in FIG. 16, the spectrum of the decaying coherence pathway may be predominantly anti-symmetric with respect to ω₀, whereas the spectrum of the recovering coherence pathway may be symmetric with respect to ω₀.

Referring to FIGS. 17A-D, there are illustrated calculated spectra and echo shapes from the pulse sequence described in expression (23). FIG. 17A illustrates the spectrum of the decaying coherence pathway for the modified sequence and FIG. 17B illustrates the spectrum of the recovering pathway. The spectra of both coherence pathways are in phase, but that of the recovering coherence pathway (see FIG. 17B) is symmetric with respect to the offset frequency, whereas that of the decaying coherence pathway (FIG. 17A) is mainly anti-symmetric. As a result, the signal in the time domain forms mainly out of phase for the recovering coherence pathway and mainly in phase for the decaying coherence pathway. The time domain echo shapes for the decaying coherence pathway and the recovering coherence pathway are illustrated in FIGS. 17C and 17D, respectively. In FIG. 17C, line 142 represents the in-phase component and line 144 represents the out-of-phase component. The smaller in-phase component in the decaying coherence pathway may be caused by the residual symmetric component in the spectrum of this coherence pathway.

To confirm the above calculations, an example was performed using the sequence described in expression (22). The echo shapes were extracted by averaging the 10^(th) to 110^(th) echo. FIGS. 18A and 18B illustrate example results of echo shapes with the modified pulse sequence according to an embodiment of the invention. In FIG. 18A, the in-phase component is illustrated and in FIG. 18B, the out-of-phase component is illustrated. The different shapes in each figure correspond to different values of recovery time, τ. As can be seen by comparing FIGS. 18A and 18B to FIGS. 17C and 17D, the shapes agree well with the calculated results. When the recovery time, τ, is increased, the out-of-phase signal decays, whereas the in-phase signal grows. The ratio of the out-of-phase to in-phase signal may be directly related to the relative size of the recovery time, τ, to the relaxation time, T₁.

In addition, the signal power of the two signals s_(r)(t) and s_(d)(t) may also be significantly increased. Referring to FIG. 19, there is illustrated a plot of signal power (represented on the vertical axis) versus a ratio of the relaxation time, T₁, to the recovery time, τ, (represented on the horizontal axis) for different pulse sequences. Line 146 represents data from an example using the standard inversion recovery CPGM sequence (illustrated in FIG. 1) operated on resonance. Line 142 represents data from an example using the modified inversion recovery CPGM sequence described by expression (23). It can be seen that the modified inversion recovery CPMG sequence may generate signals that are of comparable amplitude, or slightly larger, as those of the standard sequence. With the standard sequence (se FIG. 1), there may be substantial cancellation between the signals of the two coherence pathways when the recovery time is close to log 2 T₁. This may lead to a pronounced dip 158 in the signal power of the CPMG echoes. This problem may be avoided with the modified sequence according to an embodiment of the invention because the two contributions may be predominantly out of phase and therefore do not cancel.

The decomposition of the example echo shapes into the components of the two coherence pathways may still be described by equation (7). The amplitudes of the decay and recovery components may be extracted from the echo shapes shown in FIGS. 18A and 18B using equations (17) and (18). Referring to FIG. 20, there are illustrated the extracted amplitudes for the decaying coherence pathway (data set 154) and recovering coherence pathway (data set 156). There is excellent agreement with the expected exponential dependence on τ/T₁.

The modified sequence according to an embodiment of the invention may not generate a perfectly anti-symmetric longitudinal magnetization after the inversion pulse. As a consequence, the decaying coherence pathway may generate a small in-phase component, as shown in FIG. 17C. In one embodiment, it may be possible to extract the purely anti-symmetric signal of this coherence pathway by phase cycling. For example, referring to FIG. 16, changing the phase of the second pulse in the sequence 138 from +y to −y may effectively invert the frequency axis. In other words, this change may only affect the anti-symmetric signal. If the signals generated by the two sequences are subtracted from one another, only the anti-symmetric, out-of-phase signal may contribute. This may result in an exact subtraction of the recovering signal over the whole spectrum.

The above-described composite pulse sequence may significantly improve the orthogonality of the two signals s_(r)(t) and s_(d)(t). In one example, the normalized cross term may be:

(s _(d) ·s _(r))/√{square root over ((s _(d) ·s _(d))(s _(r) ·s _(r)))}{square root over ((s _(d) ·s _(d))(s _(r) ·s _(r)))}=0.15   (24)

This may be a significant improvement compared to the cross term value of −0.625 for the standard CPMG sequence discussed above.

With any T₁ sequence, the signal power will depend on the ratio of T₁/τ. A robust single-shot method for extraction of porosity, T₂, and T₁ may require a weak dependence in order to allow acquisition over a wide range of parameters. With the standard inversion recovery CPMG operated on resonance, the decaying and recovery signals exactly cancel for T₁=τ/log 2 and no additional information can be obtained. With the modified sequence, the dependence of the signal power on the ratio T₁/τ is much weaker, as shown in FIG. 10.

As discussed above, the amplitudes of the a_(r) and a_(d) of the component signals can be extracted from a measured echo shape S(t) according to equations (17) and (18), and T₁ ay be found from the amplitudes according to equation (19). Referring to FIG. 20, there is illustrated the relationship between the longitudinal relaxation time, T₁, and the relative sizes of the amplitudes of the decaying and coherence pathways. The ratio T₁/τ is represented on the vertical axis and the amplitude ratio from equation (19) is represented on the vertical axis. Line 150 represents theoretical results from equation (19). The measured echo shapes from the above-described example were also decomposed into a weighted sum of s_(r)(t) and s_(d)(t) using equation (16). The corresponding amplitudes, found using equations (17) and (18). In FIG. 21, the series of data points 152 represent results from the above-described example using a modified inversion recovery CPMG sequence according to an embodiment of the invention. It can be seen that the ratio of the amplitudes a_(d)/(a_(d)+a_(r)) agrees well with the prediction from equation (19). Thus, the amplitudes of the two components may be accurately described by equation (19). This example shows that it may be possible to extract T₁ from the echo shape in a single-scan measurement. For good sensitivity, FIG. 21 suggests that the recovery time, τ, may be chosen to be in the range of approximately 0.4 T₁ to 10 T₁. It is to be appreciated that is if T₁ is characterized by a distribution ƒ(T₁), (as may be the case with complex samples, such as water or oil saturated rock) then equation (19) may be replaced by:

$\begin{matrix} {\frac{a_{d}}{a_{d} + a_{r}} = \frac{\int{{T_{1}}{f\left( T_{1} \right)}\exp \left\{ {{- \tau}/T_{1}} \right\}}}{\int{{T_{1}}{f\left( T_{1} \right)}}}} & (25) \end{matrix}$

Various pulse sequences and analysis techniques according to aspects and embodiments of the invention may be used in many applications where fast T₁ measurements may be desirable. For example, such applications may include identifying the presence of light condensate and fluids with high GOR, and performing measurements in “super-k” zones in carbonate reservoirs. In some carbonate reservoirs, thin high-permeability zones may completely dominate the flow properties of a formation. Such zones may be characterized by large pores where T₁ approaches the bulk relaxation time. It has been difficult to identify these zones reliably with conventional T₂-based NMR logging approaches because of diffusion and motion effects. However, as described above, T₁ may be far less affected by such diffusion or motion effects, and thus fast, accurate measurements of T₁ according to embodiments of the invention may be useful in identifying and quantifying these zones. For example, if the formation is characterized by a bi-modal T₁ distribution with widely separated contributions f_(short)(T₁) and f_(long)(T₁), as is the case in some gas zones or super-k zones, it may be preferable to place the recovery time, τ, in the gap between the two contributions. In this case, the amplitudes a_(d) and a_(r) may correspond to the porosities associated with the short and long T₁ values, respectively. This may apply even if the T₂ distributions of the two contributions are overlapping.

In another example, the pulse sequences and analysis techniques of embodiments of the invention may be used for a single-shot T₁, T₂ and porosity measurement. For narrow relaxation time distributions, the modified inversion recovery CPMG sequence described above may be used to obtain all three quantities in a single measurement. That is, T₁ may be obtained from the echo shape, as described above, T₂ may be obtained from the decay of the echo amplitudes as known in the art, and the porosity may be obtained from the initial amplitudes. In this example, it may preferable to have the recovery time, τ, chosen between 0.4 and 10 of the expected T₁ value. In another example, the average relaxation rate, 1/T₁, may be obtained in a formation with long relaxation times by performing measurements with multiple recovery times. As discussed above, the average T₁ relaxation rate may be obtained from the amplitudes of the decaying signal, a_(d)(τ), for measurements with recovery time much less than T₁.

According to another embodiment, the above-described techniques of encoding NMR information in the echo shape may also be applied to diffusion measurements. For example, if more than two pulses are applied in grossly inhomogeneous fields, multiple echoes may form that have different sensitivities to diffusion in the static background gradient. In one embodiment, it may be preferable that at least some echoes form at nearly identical times so that CPMG detection may be used. An example of a standard diffusion editing pulse sequence is illustrated in FIG. 22. In the illustrated example, the pulse sequence comprises a CPMG sequence wherein the first two echo spacings have been increased to encode diffusion information. This type of sequence may allow the echo signal to be formed by contributions from direct and stimulated echo coherence pathways. More specifically, after the second 180° pulse 160 the direct echo and stimulated echo coherence pathways may generate an echo at the same time that may then be refocused by the subsequent long series of 180° pulses 162. This refocusing may be used to increase the signal-to-noise ratio for the echo shape measurement, as discussed above. In one example, the signal may be maximized if the first pulse spacing 166 is reduced from t_(E)/2 to t_(E)/2−t₁₈₀/π, as indicated in FIG. 22.

In one example, if the s_(de)(t) and s_(se)(t) are the echo shapes of the direct and stimulated echo coherence pathways, respectively, during the CPMG detection (in the absence of diffusion) then the measured echo shape may be given by:

$\begin{matrix} {{S(t)} \propto {{{s_{de}(t)}\exp \left\{ {{- \frac{1}{6}}\gamma^{2}g^{2}t_{E\; 1}^{3}D} \right\}} + {{s_{se}(t)}\exp \left\{ {{- \frac{1}{3}}\gamma^{2}g^{2}t_{E\; 1}^{3}D} \right\}}}} & (26) \end{matrix}$

where: g is the magnetic field gradient;

-   -   D is the diffusion coefficient; and     -   t_(E1) is the echo spacing of the first two echoes.         The weights may depend on the diffusion and relaxation         parameters of the sample, but the echo shapes the s_(de)(t) and         s_(se)(t) may depend only on the RF pulses. Referring to FIGS.         24A and 24B, there are illustrated the calculated echo shapes         S_(de)(t) and s_(se)(t) for the direct echo and stimulated echo         contributions, respectively, that correspond to the pulse         sequence of FIG. 22. The in-phase signal is shown as the solid         lines 170 a (direct echo. FIG. 24A) and 170 b (stimulated echo.         FIG. 24B), and the out-of-phase signals are shown as the dashed         lines 172 a (direct echo, FIG. 24A) and 172 b (stimulated echo,         FIG. 24B). For these calculations, it was assumed that the         nominal flip angle, θ, was 98° (θ is proportional to the         duration and strength of the RF pulse). It can be seen that for         the pulse sequence illustrated in FIG. 22, referred to herein as         Sequence A, the two contributions strongly overlap.

In one embodiment, if the two contributions can be separated, the relative amplitudes may be used to infer the diffusion coefficient from a measurement with a single encoding time. However, with the standard diffusion editing sequence, illustrated in FIG. 22, it may be difficult to separate the two components, s_(de)(t) and s_(se)(t) from each other. This may be due to the considerable overlap between the two components, as can be seen in FIGS. 24A and 24B. Therefore, according to aspects of the invention, there are provided some examples of modified pulse sequences that may allow a robust separation of the two contributions.

According to one embodiment, a pulse sequence may be used that has the form illustrated in FIG. 23. In this embodiment, the phase of either the first or second 180° pulse may be changed by 90°. Referring to FIG. 23, in the illustrated example, the phase of the second 180° pulse 164 has been shifted from y to x. As a result, the signal of the two contributions may form out of phase to each other, analogous to the methods described above in reference to the T₁ measurements. Referring to FIGS. 25A and 25B, there are illustrated calculated echo shapes for the direct echo signal (FIG. 25A) and the stimulated echo signal (FIG. 25B). In FIG. 25A, the solid line 174 a represents the in-phase signal and the dotted line 176 a represents the out-of-phase signal. Similarly, in FIG. 25B, the in-phase signal is shown as solid line 174 b and the out-of-phase signal is shown as dotted line 176 b. Again, it was assumed that θ=98°. It can be seen that with Sequence B, the contribution from the stimulated echo (FIG. 25B) form out of phase with respect to the contribution from the direct echo (FIG. 25A). Thus, Sequence B may facilitate diffusion encoding by phase separation of the signals from the two coherence pathways. In addition, the timing of the first two pulses may be slightly adjusted compared to Sequence A. In sequence B (the sequence illustrated in FIG. 23), if the pulse spacing is left identical to that of Sequence A, the contribution of the direct echo may be maximized, while the contribution of the stimulated echo may be minimized. Therefore, in one example, the timing of Sequence B may be adjusted such that the first pulse spacing 168 may be t_(E)/2, as shown. This may greatly increase the stimulated echo contribution and only modestly decrease the direct echo signal.

An example was performed to test Sequence B on a sample of doped water. In this example, the initial echo spacing, t_(E1), was systematically increased in 32 steps to a maximum value of 17.1 ms. For each case, the CPMG detection comprises 2000 echoes that were acquired with an echo spacing of t_(E1)=424 μs. The durations of the nominal 90° pulse 178 and 180° pulses 180 and 164 were t₉₀=24 μs and were t₁₈₀=48 μs. Referring to FIGS. 26A-26E, there are illustrate the in-phase and out-of-phase components of measured echo shapes, S(t), for phase encoding of diffusion information using Sequence B. A series of five examples were performed with different values of the factor γ²g²t³ _(E1)D. FIGS. 26A-E illustrate the echo shapes during CPMG detection for each of these five different values. The echo shapes were extracted from the acquired data by averaging the shapes of the 100^(th) to 300^(th) echo. FIG. 26A shows the extracted echo shape for γ²g²t³ _(E1)D=0.01. Line 182 a represents the in-phase signal and line 184 a represents the out-of-phase signal. In FIG. 26B, the extracted echo shape for γ²g²t³ _(E1)D=0.4 is presented, with the in-phase signal shown as line 182 b and the out-of-phase signal shown as line 184 b. Similarly, in FIG. 26C, the echo shape for γ²g²t³ _(E1)D=3.1 is presented, with the in-phase signal shown as line 182 c and the out-of-phase signal shown as line 184 c. FIG. 26D illustrates the echo shape for γ²g²t³ _(E1)D=4.4. Line 182 d represents the in-phase signal and line 184 d represents the out-of-phase signal. Lastly, FIG. 26E illustrates the echo shape for γ²g²t³ _(E1)D=6.5, with line 182 e representing the in-phase signal and line 184 e representing the out-of-phase signal. It can be seen that, in agreement with the theoretical expectation illustrated in FIGS. 25A and 25B, there are both sizable in-phase and out-of-phase signals. The in-phase signals (lines 182 a-e) are symmetric and are generated by the direct echo coherence pathways, while the out-of-phase signals (lines 184 a-e) are anti-symmetric and are generated by the stimulated echo coherence pathways. As the initial echo spacing is increased, diffusion may become more important, and may preferentially reduce the out-of-phase component.

In one example, for each echo, the amplitudes of the two components may be extracted using as matched filters the expected echo shapes in the absence of diffusion. Extracted amplitudes, corresponding to the echo shapes shown in FIGS. 26A-E, are illustrated in FIGS. 27A-E versus time after the initial 90° pulse 178. Within the CPMG train, the echo shapes remain unchanged, as discussed above, and the amplitudes of all components may decay with substantially the same relaxation time. In FIGS. 27A-27E, amplitudes of the in-phase components are shown as series 186 a-e, respectively, and the amplitudes of the out-of-phase signals are shown as series 188 a-e, respectively. In FIGS. 27A and 27B, the two series are very close to one another and cannot be easily distinguished. However, the separation increases in FIGS. 27C-E. In each case, the solid lines 190 and 192 are an exponential fit to single exponential decays for the in-phase and out-of-phase components, respectively. Substantially identical transverse relaxation times, T_(2,eff)=114 ms, may be found for all measurements.

Referring to FIG. 28, there is illustrated a plot of the fitted initial amplitudes, extrapolated to zero time, for the in-phase and out-of-phase components versus the dimensionless diffusion coefficient, γ²g²t³ _(E1)D. The initial amplitudes of the in-phase contribution are shown as series 194 and the initial amplitudes of the out-of-phase contribution are shown as series 196. The solid line 198 shows the expected dependence for the direct echo coherence pathway and the solid line 200 shows the expected dependence for the stimulated echo coherence pathway. It can be seen from FIG. 28 that the amplitudes of both the in-phase and out-of-phase components may decay exponentially as a function of the dimensionless diffusion coefficient, γ²g²t³ _(E1)D, but with exponents that differ by a factor of 2. This confirms that the in-phase and out-of-phase components are associated with the direct echo or stimulated echo coherence pathways, respectively. In addition, there is excellent agreement between the example results and the theoretical prediction of equation (26) (represented by lines 198 and 200). It should be noted that the amplitudes of the first few echoes show a characteristic transient effect, similar to that observed with a standard CPMG sequence. This is well understood in the art and is caused by imperfect averaging of the magnetization perpendicular to the net axis, {circumflex over (n)}. As discussed above, these effects may affect only the first few echoes and therefore, for the later echoes, the asymptotic expression in equation (2) may be an excellent approximation. Also, for the higher values of the diffusion coefficient, γ²g²t³ _(E1)D, it can be seen that the measured amplitudes 196 of the out-of-phase component are somewhat above the theoretical prediction of line 200. This may be due to a small mixing of the direct echo coherence pathway into the out-of-phase signal. This mixing may be caused by asymmetries of the sample shape along the field gradient, asymmetry in the frequency response of the NMR detection system, or imperfect tuning.

According to another embodiment, a pulse sequence may be used to implement diffusion encoding by separating the contributions of the two coherence pathways temporally. One example of such a pulse sequence, referred to herein as Sequence C, is illustrated in FIG. 29. In one example, the two contributions (the direct echo and stimulated echo coherence pathways) may be separated directly in the time domain by increasing the second echo spacing 202 by a small amount, τ_(c). The stimulated echo may then form a duration, τ_(c), earlier than the direct echo. The echoes in the CPMG detection may then comprise a central peak 204 (illustrated in FIG. 30A) due to the direct echo and two symmetric side peaks 206 and 208 (see FIG. 30B) due to the stimulated echo. The two side peaks may be shifted by ±(τ_(c)−2t₁₈₀/π). The value of τ_(c) may be large enough to separate the two contributions (i.e., larger than t₁₈₀), but small enough so that the shifted peaks still lie within the detection window (i.e., less than t_(E)/2). In this calculation, the extra time delay, τ_(c), was 2.5 t₁₈₀. The nominal flip angle, θ, may control the relative weights of the two coherence pathways, and was assumed, for this calculation, to be 98°.

An example was performed to test diffusion encoding by separating the contributions of the two coherence pathways in time using Sequence C. In this example, the durations of the nominal 90° pulse 178 and 180° pulses 210 and 212 were t₉₀=12 μs and were t₁₈₀=24 μs. The duration of the second and third pulses 210, 212 was set to 14 μs, corresponding to a nutation angle, θ=105° on resonance. The delay, τ_(c), was set to 60 μs=2.5 t₁₈₀. Data was acquired for 32 different initial echo spacing, t_(E1), up to a maximum of 17.1 ms. The CPMG detection comprises 2000 echoes, acquired with an echo spacing of t_(E)=400 μs.

Referring to FIGS. 31A-E, there are illustrated results from this example. Again, five examples with different diffusion coefficients (i.e., different initial echo spacing) were performed. In each of FIGS. 31A-E, the in phase components of the echo shape, S(t), are illustrated as a function of time. FIG. 31A shows the echo shape for γ²g²t³ _(E1)D=0.01, FIG. 31B shows the echo shape for γ²g²t³ _(E1)D=0.4, FIG. 31C shows the echo shape for γ²g²t³ _(E1)D=3.1, FIG. 31D shows the echo shape for γ²g²t³ _(E1)D=4.5, and FIG. 31E shows the echo shape for γ²g²t³ _(E1)D=6.7. As can be seen in the figures, the echoes form in phase with a central peak 214 and two symmetrical satellite peaks 216. When the diffusion constant is small (i.e., diffusion is not an important effect), the amplitudes of the central peak and the two satellite peaks are very similar. As the initial echo spacing is increased, corresponding to a larger value of the diffusion coefficient (and the progression from FIG. 31A to FIG. 31E), the satellite peaks, which are due to stimulated echo coherence pathway, attenuate more than does the central peak which is generated by the direct echo coherence pathway.

As discussed above in reference to Sequence B, the amplitudes of the two components may be extracted using the expected echo shapes for the two pathways as matched filters. From the decay of these amplitudes, the transverse relaxation time, T_(2,eff) may be extracted, as discussed above. For this example, a value of T_(2,eff)=114 ms was obtained, the same as for the example using Sequence B. Referring to FIG. 32, there is illustrated a plot of the fitted amplitudes of the two contributions versus the dimensionless diffusion coefficient, γ²g²t³ _(E1)D for diffusion encoding by temporal separation of the two contributions. The fitted initial amplitudes of the central peak are represented by data series 218 and the fitted initial amplitude of the satellite peaks are represented by data series 220. Line 222 is an expected dependence of the direct echo coherence pathway, and line 224 is the expected dependence of the stimulated echo coherence pathway. It can be seen that there is excellent agreement between the example results and the theoretical predictions (lines 222 and 224), even at high values of the diffusion coefficient (corresponding to large diffusion effects).

These examples illustrate that for both implementations of diffusion encoding in the echo shape (i.e., by phase separation or time separation), the value of the diffusion coefficient may be extracted from the measurement with a single value of t_(E1). The results shown in FIGS. 28 and 32 demonstrate that the ratio of the amplitudes of the two components may be directly related to the diffusion coefficient. It is to be appreciated that the principles of the invention are not limited to the example pulse sequences described and illustrated herein, and other pulse sequences may also be used to measure diffusion, relaxation, and other parameters. For example, in samples where T₁ is not equal to T₂, some relaxation effects may be present during the encoding sequence that may have an effect on the diffusion coefficient measurement. Therefore, in another example, a pulse sequence may be used to compensate for such relaxation effects by choosing two coherence pathways that have identical relaxation, but different diffusion sensitivities.

Aspects of the invention described herein in reference to a single-shot T₁ measurement may also be extended to other applications. For example, any of the diffusion sequences described herein (e.g., Sequences A, B or C) may also be used for the detection and quantification of flow or convection. In one example, to the first order, the direct echo coherence pathway may be unaffected by flow, whereas the in-phase amplitude of the stimulated echo coherence pathway may be sensitive to the mean squared displacement of the spins with respect to the applied magnetic fields. When coherent motion dominates, this measurement may provide a single-shot determination of the magnitude of the average flow velocity.

According to various aspects and embodiments of the invention, it may be possible to encode information in grossly inhomogeneous fields not only in the amplitude, but also in the shape of the echoes in a CPMG train. This may allow a variety of measurements, including single-shot measurements of diffusion and T₁. Having thus described several aspects and embodiments of the invention, modifications and/or improvements may be apparent to those skilled in the art and are intended to be part of this disclosure. For example, it is to be understood that the invention may not require the use of CPMG detection, and other detection sequences may be used, such as a free induction decay sequence or a spin echo sequence. It is to be appreciated that the invention is not limited to the specific examples described herein and that the principles of the invention may be applied to a wide variety applications. The above description is therefore by way of example only, and includes any modifications and improvements that may be apparent to one of skill in the art. The scope of the invention should be determined from proper construction of the appended claims and their equivalents. 

1. A method of measuring a longitudinal relaxation time in a sample having an initial magnetization, the method comprising: disturbing the initial magnetization with a first series of RF pulses; after a recovery time period has elapsed, applying a second series of RF pulses to the sample to acquire a first signal comprising at least two echoes; disturbing the initial magnetization differently with a third series of RF pulses; after the recovery time period has elapsed, applying a fourth series of RF pulses to the sample to acquire a second signal comprising at least two echoes; obtaining a difference signal from the first signal and the second signal; and analyzing the difference signal to obtain the longitudinal relaxation time.
 2. The method as claimed in claim 1, wherein applying a fourth series of RF pulses comprises reapplying the second series of RF pulses.
 3. The method as claimed in claim 2, wherein the second series of RF pulses comprises a Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence.
 4. The method as claimed in claim 1, wherein analyzing the difference signal comprises fitting a function to the difference signal.
 5. The method as claimed in claim 4, wherein the function comprises one of a single exponential function, a double exponential function, a one-dimensional numerical Laplace inversion, and a two-dimensional Laplace inversion.
 6. The method as claimed in claim 1, wherein the first series of RF pulses comprises a first pair of 90° pulses and the third series of RF pulses comprises a second pair of 90° pulses, and wherein the first and second pairs of 90° RF pulses have different phase cycling.
 7. The method as claimed in claim 1, wherein first series of RF pulses comprises a 180° pulse and wherein the third series of RF pulses comprises no corresponding pulse.
 8. The method as claimed in claim 1, further comprising repeating the steps of disturbing the initial magnetization, applying the second series of RF pulses, disturbing the initial magnetization differently, and applying the fourth series of RF pulses, for a series of values of the recovery time period.
 9. The method as claimed in claim 8, wherein analyzing the difference signal includes analyzing the difference signal to determine decay of the difference signal for different echoes and different recovery times to obtain the longitudinal relaxation time.
 10. A method of measuring a longitudinal relaxation time in a sample having an initial magnetization, the method comprising: applying a sequence of RF pulses to the sample, the sequence including an encoding portion and a detection portion; acquiring an echo signal using the detection portion of the sequence of RF pulses; decomposing the echo signal into at least two coherence pathway components; and analyzing at least one of the two coherence pathway components to determine the longitudinal relaxation time.
 11. The method as claimed in claim 10, wherein decomposing the echo signal into at least two coherence pathway components includes decomposing the echo signal into a decay component and a recovery component.
 12. The method as claimed in claim 11, wherein analyzing at least one of the two coherence pathway components includes analyzing the decay component.
 13. The method as claimed in claim 11 wherein the encoding portion of the sequence of RF pulses comprises a pair of 127° pulses separated from one another by a first time period.
 14. A method of measuring diffusion in a sample, the method comprising: applying a sequence of RF pulses to the sample, the sequence including an encoding portion and a detection portion; acquiring an echo signal using the detection portion of the sequence of RF pulses; decomposing the echo signal into at least two coherence pathway components; and analyzing the at least two coherence pathway components to determine a diffusion coefficient.
 15. The method as claimed in claim 14, wherein decomposing the echo signal into at least two coherence pathway components includes decomposing the echo signal into a direct echo component and a stimulated echo component.
 16. The method as claimed in claim 15, wherein analyzing the at least two coherence pathway components includes: extracting a first amplitude of the direct echo component; extracting a second amplitude of the stimulated echo component; and determining the diffusion coefficient from a ratio of the first and second amplitudes.
 17. The method as claimed in claim 15, wherein the detection portion of the sequence of RF pulses comprises a Carr-Purcell-Meiboom-Gill (CPMG) pulse train.
 18. The method as claimed in claim 15, wherein the encoding portion of the sequence of RF pulses comprises a pair of 180° pulses having phases that differ by 90°.
 19. A nuclear magnetic resonance measurement device comprising: a transmitter constructed and arranged to generate a sequence of RF pulses and to apply the sequence of RF pulses to a sample; a receiver constructed and arranged to receive an echo signal from the sample; and a processor constructed and arranged to decompose the echo signal into at least two coherence pathway components, and to analyze at least one of the two coherence pathway components to determine a parameter of the sample.
 20. The nuclear magnetic resonance measurement device as claimed in claim 19, wherein the sequence of RF pulses comprises an encoding portion and a detection portion.
 21. The nuclear magnetic resonance measurement device as claimed in claim 20, wherein the detection portion comprises a Carr-Purcell-Meiboom-Gill (CPMG) pulse train.
 22. The nuclear magnetic resonance measurement device as claimed in claim 21, wherein the encoding portion of the sequence of RF pulses comprises a pair of 127° pulses separated from one another by a first time period
 23. The nuclear magnetic resonance measurement device as claimed in claim 21, wherein the encoding portion of the sequence of RF pulses comprises a pair of 180° pulses having phases that differ by 90°
 24. The nuclear magnetic resonance measurement device as claimed in claim 19, wherein the processor is constructed and arranged to decompose the echo signal into a decay component and a recovery component.
 25. The nuclear magnetic resonance measurement device as claimed in claim 24, wherein the parameter of the sample is a longitudinal relaxation time of the sample, and wherein the processor is constructed and arranged to analyze the decay component to obtain a measurement of the longitudinal relaxation time of the sample.
 26. The nuclear magnetic resonance measurement device as claimed in claim 19, wherein the processor is constructed and arranged to decompose the echo signal into a direct echo component and a stimulated echo component.
 27. The nuclear magnetic resonance measurement device as claimed in claim 26, wherein the parameter of the sample is a diffusion coefficient, and wherein the processor is constructed and arranged to analyze the direct echo component and the stimulated echo component to determine the diffusion coefficient.
 28. The nuclear magnetic resonance measurement device as claimed in claim 27, wherein the processor is constructed and arranged to analyze the direct echo component to extract a first amplitude, and to analyze the stimulated echo component to extract a second amplitude of the stimulated echo component; and wherein the processor is configured to determine the diffusion coefficient from a ratio of the first and second amplitudes. 